Latent Probability: Conditional Dependence as Graded Spectral Structure

We embed conditional probability in the graded Hilbert tensor algebra \(\Lambda^{(i,j)}\) of the Latent framework. The conditional distribution \(P(X \mid Y)\) is identified with a grade-2 linear map \(\Lambda^{(X,Y)}: \mathcal{H}_Y \to \mathcal{H}_X\) between latent spaces. A Bayesian network is the sparsity pattern of the grade-2 tensor. We prove:

1. Markov Emergence Theorem. If the grade-2 tensor of a system with \(n\) variables has meta-analyticity \(\rho_\mathrm{meta} > 1\), then the induced probability structure satisfies approximate conditional independence with Markov blanket size at most \(R = O(\log(n/\varepsilon) / \log \rho_\mathrm{meta})\). The Markov property is a consequence of rank deficiency, not an axiom.

2. Dependency Classification Theorem. Every dependency cycle in a graded system is classified by the monodromy spectrum of the composed grade-2 maps. Let \(M = \Lambda^{(A,C)} \circ \Lambda^{(C,B)} \circ \Lambda^{(B,A)}\) be the monodromy of a cycle \(A \to B \to C \to A\). Then: - If \(|\mu_i| < 1\) for all eigenvalues: the cycle is transient — it dissolves at grade \(n+1\). - If \(|\mu_i| = 1\) for some eigenvalues: the cycle is resonant — it becomes a symmetry at grade \(n+1\). - If \(|\mu_i| > 1\) for some eigenvalues: the cycle is unstable — it requires grade \(n+1\) nonlinearity to regularize.

3. Graphical Model Subsumption. The Hammersley-Clifford theorem, Pearl's d-separation criterion, and the factorization theorem for Bayesian networks are special cases of the grade-2 spectral decomposition.

The framework connects probability theory to dynamical systems: periodic orbits in the \(N\)-body problem are resonant dependency cycles whose monodromy is exactly the Poincaré return map. The virial crossing property of bounded orbits (Lagrange-Jacobi) is the constraint that prevents unstable cycles from persisting in bounded systems.